Integration - Building Totals from Rates of Change

Integration is the reverse of differentiation. If differentiation breaks things apart to find rates of change, integration builds them back together to find totals – areas, volumes, accumulated quantities. Integration is one of the most powerful tools in all of mathematics and science.

The elongated “S” symbol ∫ was introduced by Gottfried Wilhelm Leibniz in 1675, standing for the Latin word summa (“sum”), because he thought of integration as adding up infinitely many infinitely thin slices. The link between integration and differentiation – now called the Fundamental Theorem of Calculus – was actually glimpsed before either Newton or Leibniz: Newton's own teacher at Cambridge, Isaac Barrow, came remarkably close to stating it in his 1670 geometry lectures, and the Scottish mathematician James Gregory proved a special case of it two years earlier still. It took Newton and Leibniz to recognise the theorem's full generality and turn it into a systematic method, which is why they, rather than Barrow or Gregory, are remembered as calculus's inventors.

Two Types of Integration

Indefinite integrals find a general family of functions (always including + C).
Definite integrals calculate a specific numerical value – the net area between a curve and the x-axis over a given interval.

The Antiderivative

The integral of f(x) is a function F(x) such that F′(x) = f(x). It is called the antiderivative.
Notation: ∫ f(x) dx = F(x) + C
The constant C appears because many functions share the same derivative – for example, x², x²+5, and x²−100 all differentiate to 2x.

Basic Integration Rules

Function f(x)	Integral ∫f(x) dx
xⁿ (n ≠ −1)	xⁿ⁺¹ / (n+1) + C
1/x	ln\|x\| + C
e^x	e^x + C
sin x	−cos x + C
cos x	sin x + C
sec² x	tan x + C
k (constant)	kx + C

The Fundamental Theorem of Calculus

This theorem is the bridge between differentiation and integration. It has two parts:
Part 1: If F(x) = ∫_a^x f(t) dt, then F′(x) = f(x). Differentiation undoes integration.
Part 2: ∫_a^b f(x) dx = F(b) − F(a), where F is any antiderivative of f.
Part 2 is the practical rule you use to evaluate definite integrals.

Worked Examples

Find ∫ (3x² + 4x − 5) dx.

Integrate term by term:
∫ 3x² dx = x³, ∫ 4x dx = 2x², ∫ −5 dx = −5x.
Answer: x³ + 2x² − 5x + C.

Evaluate ∫₁³ (2x + 1) dx.

Antiderivative: F(x) = x² + x.
F(3) − F(1) = (9 + 3) − (1 + 1) = 12 − 2 = 10.

Find the area under y = x² between x = 0 and x = 3.

∫₀³ x² dx = [x³/3]₀³ = 27/3 − 0 = 9 square units.

Find ∫ sin(2x) dx.

Use reverse chain rule. The antiderivative of sin(u) is −cos(u); the inner derivative of 2x is 2.
∫ sin(2x) dx = −cos(2x)/2 + C.

Integration by Substitution

When an integrand contains a composite function, substitute u = g(x) to simplify it.
Steps: 1. Choose u = inner function. 2. Find du/dx and write dx = du / g′(x). 3. Rewrite the integral in terms of u. 4. Integrate. 5. Back-substitute x.

Find ∫ 2x(x² + 1)⁴ dx.

Let u = x² + 1. Then du = 2x dx.
∫ u⁴ du = u⁵/5 + C = (x²+1)⁵/5 + C.

Key Takeaways

Integration is the reverse of differentiation.
Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1).
Definite integral = F(b) − F(a) = exact area under the curve from a to b.
Always add + C to indefinite integrals.

Practice: Evaluating Integrals

Evaluating a Definite Integral